Hedging market risk in optimal liquidation

Phillip Monin

The Office of Financial ResearchWashington, DC

Conference on Stochastic Asymptotics & Applications,Joint with 6th Western Conference on Mathematical Finance

And to honor Jean-Pierre Fouque on the occasion of his birthday

University of California, Santa BarbaraSeptember 2014

Views and opinions are those of the speaker and do not necessarilyrepresent official OFR positions or policy.

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I Established by Dodd-Frank ’10 to support the Financial StabilityOversight Council (a senior risk management committee for thefinancial system), its member agencies, and the public:

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promoting data integrity, accuracy, and transparency for the benefitof market participants, regulators, and research communities

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J.P. Fouque: member of NIF, current member of OFR’s FRAC

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I Fundamental research:

I Hedging Market Risk in Optimal LiquidationI On the Optimal Wealth Process...I Trust and Stability in a Financial NetworkI Crisis GreeksI Asset Managers with Multiple Heterogeneous ClientsI Contingent Claim AnalysisI Polysplines for Portfolio Payoffs

Hedging market risk in optimal liquidation

I Motivation and background

I Market model and terminal portfolio value

I Solution for CARA utility

I Analysis and plots of optimal strategies

I Broker-dealer’s minimum spread

Motivation

Standard models assume markets are perfectly liquid.

Investors are often liquidity demanders.

Instantaneous liquidity often not available or expensive:

I Find a dark pool (recent talk of increased regulation)

I Pay a broker-dealer’s block/capital markets desk WSJ

I Standard solution: break it up and liquidate over time

Liquidating over time involves market risk.

Take symmetric position in correlated but relatively liquid asset.

Is this behavior “optimal”? What is the hedge?

Setup

Investor must liquidate a large long position in a primary asset over [0,T ].

Investor is liquidity demander; no inside information.

Proceeds deposited into a riskfree money market account.

Investor trades between money market and liquid proxy.

Investor maximizes expected terminal utility.

Preview of results:

I Almgren-Chriss type model with liquid proxy.

I Optimal strategies deterministic, found explicitly.

I Hedge for market risk effectively makes for more aggressiveliquidation.

I Indifference price (spread) for broker-dealer trading as principal.

I Always better to find and trade in liquid proxy.

Background

Numerous studies (Kraus & Stoll (’72), Holthausen et al (’87,’90), Keim& Madhavan (’95), Almgren et al (’05), Frino et al (’06), etc.)

I Temporary price impact

I Permanent price impact

Many microstructure models (Kyle (’85), Easley & O’Hara (’87), etc.)attempt to explain these endogenously.

A separate line takes price impact effects as exogenous and then derivesoptimal strategies.

Most popular is Almgren-Chriss (’99,’00,’03) model:

I Incorporates temporary and permanent price impacts.

I Mathematically tractable.

I Widely used in academia and practice.

Almgren-Chriss type market environment

Over the horizon [0,T ], the market consists of

I Riskless money market that pays no interest;

I Proxy with price St given by Bachelier model with drift:

St = S0 + µt + σWt

I Primary asset with price S It following simple Almgren-Chriss model:

S It = S I

0 + µI t + σIWIt︸ ︷︷ ︸

“unaffected” component

+ γ(ηt − η0)− θξt︸ ︷︷ ︸impact component

I ηt , number of shares at time t (absolutely continuous),I ξt , speed of liquidation, i.e. ηt = η0 −

∫ t

0ξudu (uniformly bounded),

I γ ≥ 0, coefficient of permanent price impact,I θ > 0, coefficient of temporary price impact,I d

⟨W ,W I

⟩t

= ρ dt, ρ ∈ [0, 1).

Portfolio value

1. Initial money market account value is zero.

2. Liquidation: investor sells ξtdt shares at time t for price S It

I Value at time t is∫ t

0ξsS

Is ds.

3. Form portfolio with money market and proxy.

Value at terminal time T , using ηT = 0:

Xπ,ξT = x0 + µI

∫ T

0

ηsds + σI

∫ T

0

ηsdWIs − θ

∫ T

0

ξ2s ds

+µ

∫ T

0

πsds + σ

∫ T

0

πsdWs ,

whereI πt , number of shares in proxy asset at time t (uniformly bounded),I x0 := S I

0η0 − γ2η2

0 .

Regularity

Solution for exponential utility

The investor’s objective is to find the policy (π, ξ) ∈ A that solves

sup(π,ξ)∈A

E[− exp(αXπ,ξT )], α > 0.

Conjecture: optimal strategy is deterministic

Intuition: optimal strategies for CARA investor are deterministic in:

I standard Merton model for optimal investment,

I “pure” liquidation model (Schied, Schoneborn & Tehranchi (’10)).

TheoremLet the positive constant κρ be defined by

κρ :=

√ασ2

I (1− ρ2)

2θ.

Then, the investor’s unique optimal policy is the deterministic strategy(π∗, ξ∗) given by

π∗t =1

α

µ

σ2− ρσI

ση∗t ,

and

ξ∗t = κρη0cosh(κρ(T − t))

sinh(κρT )+

(ρµ

σ− µI

σI

)eκρ(T−t) − eκρt√

2αθ(1− ρ2)(eκρT + 1),

where

η∗t = η0sinh(κρ(T − t))

sinh(κρT )−(ρµ

σ− µI

σI

)(eκρ(T−t) − 1)(eκρt − 1)

α(1− ρ2)σI (eκρT + 1).

Proof DPP

Broker-dealer’s indifference price

PropositionThe investor’s indifference price h(η0, 0) at time t = 0 is given by

h(η0, 0) =γ

2η2

0 + θ

∫ T

0

(ξ∗t )2dt +α

2(1− ρ2)σ2

I

∫ T

0

(η∗t )2dt

−

(S I

0η0 +1

2α

µ2

σ2T + µI (1− ρ)

∫ T

0

η∗t dt

).

The no-drift market, µ = µI = 0

0 2 4 6 8 10

Time t

0

5

10

15

20

25

30

35

40S

pee

dof

liqu

idati

onξ∗ t

Speed of Liquidation

ρ = 0

ρ = 0.50

ρ = 0.75

ρ = 0.99

Figure: Model parameters:α = 10, σ = 0.03 = σI = 0.03, γ = 0.3, θ = 0.05, η0 = 10,S0

I = 1.5. Median6-mo correlation S&P 500: 0.55, CBOE Avg Imp Corr Index (Jan 14): 0.55.

The no-drift market, µ = µI = 0

0 2 4 6 8 10

Time t

−100

−80

−60

−40

−20

0

Posi

tion

inp

roxy

ass

etπ∗ t

Position in Proxy Asset

ρ = 0

ρ = 0.50

ρ = 0.75

ρ = 0.99

Figure: Model parameters:α = 10, σ = 0.03 = σI = 0.03, γ = 0.3, θ = 0.05, η0 = 10,S0

I = 1.5. Median6-mo correlation S&P 500: 0.55, CBOE Avg Imp Corr Index (Jan 14): 0.55.

The no-drift market, µ = µI = 0

0 2 4 6 8 10

Time t

0

20

40

60

80

100

Posi

tion

inp

rim

ary

ass

etη∗ t

Position in Primary Asset

ρ = 0

ρ = 0.50

ρ = 0.75

ρ = 0.99

Figure: Model parameters:α = 10, σ = 0.03 = σI = 0.03, γ = 0.3, θ = 0.05, η0 = 10,S0

I = 1.5. Median6-mo correlation S&P 500: 0.55, CBOE Avg Imp Corr Index (Jan 14): 0.55.

PropositionThe following assertions hold in the no-drift market:i) We have

limρ↑1

ξ∗t =η0

T, 0 ≤ t ≤ T .

ii) Holding all other parameters fixed, we have that

ξ∗t (α, ρ) = ξ∗t (α, ρ) and η∗t (α, ρ) = η∗t (α, ρ)

for all t ∈ [0,T ], if and only if

α(1− ρ2) = α(1− ρ2).

iii) Holding all other parameters fixed, we have that

ξ∗t (θ, ρ) = ξ∗t (θ, ρ) and η∗t (θ, ρ) = η∗t (θ, ρ)

for all t ∈ [0,T ], if and only if

1− ρ2

θ=

1− ρ2

θ.

PropositionThe following assertions hold in the no-drift market:

i) The investor’s value function at initial time is

v(0, x0, η0) = − exp(−αx0 + αθκρ coth(κρT )η2

0

),

where x0 = S I0η0 − γ

2 η20 .

ii) The investor’s value function is increasing in the correlation ρ.

iii) The investor’s indifference price at initial time t = 0 is given by

h(η0, 0) =(γ

2+ θκρ coth(κρT )

)η2

0 − S I0η0.

0.0 0.2 0.4 0.6 0.8 1.0

Correlation ρ

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Per

centa

ge

of

book

valu

eBroker-dealer’s Minimum Spread

Conclusions and future directions

I Almgren-Chriss type model with liquid proxy.

I Optimal strategies deterministic, found explicitly.

I Hedge for market risk effectively makes for more aggressiveliquidation.

I Indifference price for broker-dealer trading as principal.

I Always better to find and trade in liquid proxy.

Future directions

I Endogenous liquidation time. Does presence of proxy increaseliquidation horizon? Recent work by Bechler & Ludkovski (’14).

I More general models and risk criteria. Several large traders.

Thank you!

Source: “Banks’ Booming Business in Block Trades Faces New Risk”,M. Jarzemsky, WSJ, 17 April 2014 Return

RegularityRegularity: optimal strategies for reasonable criteria exist and arewell-behaved.

Formed in terms of expected revenues in no-drift market.

Three popular notions. Absence of

I Price manipulation (Huberman & Stanzl (’04))

I Transaction-triggered price manipulation (Alfonsi et al (’12))

I Negative expected execution costs (Klock et al (’12))

Our model satisfies all three regularity conditions.

Expected terminal wealth (no-drift): E[Xπ,ξT ] = x0 − θ

∫ T

0ξ2t dt.

Jensen’s inequality implies ξ∗t = η0

T .

So-called VWAP (Volume-Weighted Average Price) strategy:

Xπ∗,ξ∗

T = η0

(1T

∫ T

0S It dt)

.

Return

Strategy and sketch of proof

1. Show

sup(π,ξ)∈A

E[− exp(αXπ,ξT )] = sup

(π,ξ)∈Adet

E[− exp(αXπ,ξT )].

2. Calculus of variations to find unique maximizer (π∗, ξ∗) over Adet.

3. Uniqueness follows from strict concavity of (π, ξ) 7→ E[u(Xπ,ξT )].

Sketch of proof: Write, for general (π, ξ) ∈ A,

E[u(Xπ,ξT )] = −e−αx0E

[eY

π,ξT +f (π,ξ)

],

where

Y π,ξT = −α

(σ

∫ T

0

πtdWt +

∫ T

0

ηtσI [ρdWt +√

1− ρ2dW⊥t ]

),

f (π, ξ) = −α

(µ

∫ T

0

πtdt + µI

∫ T

0

ηtdt − θ∫ T

0

ξ2t dt

).

Then, for general (π, ξ) ∈ A,

E[u(Xπ,ξT )] = = −e−αx0E

[eY

π,ξT +f (π,ξ)

]= −e−αx0E

eYπ,ξT − 1

2 〈Yπ,ξ〉T︸ ︷︷ ︸Radon-Nikodym derivative?

· e12 〈Yπ,ξ〉T +f (π,ξ)

Then, for general (π, ξ) ∈ A,

E[u(Xπ,ξT )] = = −e−αx0E

[eY

π,ξT +f (π,ξ)

]= −e−αx0E

[eY

π,ξT − 1

2 〈Yπ,ξ〉T e12 〈Yπ,ξ〉T +f (π,ξ)

]= −e−αx0Eπ,ξ

[e

12 〈Yπ,ξ〉T +f (π,ξ)

]≤ −e−εe−αx0Eπ,ξ

[e

12 〈Yπ

ε,ξε〉T

+f (πε,ξε)]

= −e−εe−αx0+ 12 〈Yπ

ε,ξε〉T

+f (πε,ξε)

= e−εE[u(Xπε,ξε

T )].

Maximizing over (π, ξ) ∈ Adet is equivalent to minimizing∫ T

0

F (t, y(t), y ′(t))dt, y(t) = (πt , ηt),

for some specific F , over curves with y(0) = (0, η0) and y(T ) = (π, 0).

Euler-Lagrange and strict convexity imply that the unique solution is thesolution to the second-order ODE

2θηt − (1− ρ2)ασ2I ηt =

µ

σρσI − µI ,

with boundary conditions

η0 = η0 > 0, ηT = 0.

Return

No need for dynamic programming or HJB equations

Heuristic arguments suggest that the value function, defined by

v(T − t, x , η) := sup(π,ξ)∈A

E[u(Xπ,ξT )|Xπ,ξ

t = x , ηξt = η],

satisfies the Hamilton-Jacobi-Bellman equation,

vt =1

2σ2I η

2vxx + µIηvx

+ sup(π,ξ)∈A

[(µvx + ρσσIηvxx)π +

1

2σ2vxxπ

2 − vηξ − θvxξ2

]= 0,

subject to the terminal condition

limt↑T

v(t, x , η) =

{−e−αx , η = 0−∞, otherwise.

One shows that the our value function is a smooth solution to the HJB.Return